File size: 12,242 Bytes
59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 | import OAI.Geometry.IsometricImmersion.Immersions.HeightNumericParameters
import OAI.Geometry.IsometricImmersion.Energy.DirectedParameterChoice
import OAI.Geometry.IsometricImmersion.Energy.PatchDirectedCoercivity
namespace OAI
noncomputable section
open Set Function
open scoped ContDiff
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model
theorem exists_patch_uniform_numeric_directed_flow
{g0 eta : MetricField} {z : Coord → ℝ} {U : Set Coord} {G Z d c e0 kappa : ℝ}
(hg : SmoothPositiveOn (g0 + eta) U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hz : ContDiffOn ℝ ∞ z U) (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) (he0 : 0 < e0)
(hgB : ∀ i j : Fin 2, CoordinateBound (fun p => (g0 + eta) p i j) modelSquare 4 G)
(hzB : CoordinateBound z modelSquare 5 Z)
(hdet : ∀ p ∈ modelSquare, d ≤ |((g0 + eta) p).det|)
(hyy : ∀ p ∈ modelSquare, c ≤ |covHessian (g0 + eta) z p 1 1|)
(hEfloor : ∀ p ∈ modelSquare, e0 ≤ heightEnergy (g0 + eta) z p)
(hsmall : ∀ p ∈ modelSquare, |hessianQuotient (g0 + eta) z p| ≤ (1 : ℝ) / 100)
(hD : ∀ p ∈ modelSquare,
(covHessian (g0 + eta) z p).det = gaussianCurvature (g0 + eta) p * heightEnergy (g0 + eta) z p)
(hk : 0 < kappa)
(hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p)
(hsupport : tsupport eta ⊆ patchBox)
(hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2)
(ell : ℕ) :
let epsilon := heightDirectedEpsilon G Z d c e0 kappa ell
let lambda := heightDirectedLambda G Z d c e0 kappa ell
let c1 := heightDirectedSlope G Z d c e0 kappa ell
let c2 := heightDirectedCoercivity G Z d c e0 kappa ell
∃ Y : ℝ → ℝ → ℝ,
0 < epsilon ∧ epsilon ≤ 1 ∧ 0 < lambda ∧ 0 < c1 ∧ 0 < c2 ∧
ContDiffOn ℝ ∞ (capChart Y) capChartDomain ∧
ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2) ∧
(∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-hessianQuotient (g0 + eta) z (coordinatePoint t (Y s t)))
(Icc (-2 : ℝ) 2) t) ∧
(∀ p ∈ capChartDomain, |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50) ∧
ContDiffOn ℝ ∞ (heightChartA (g0 + eta) z Y) capChartDomain ∧
ContDiffOn ℝ ∞ (heightChartB (g0 + eta) z Y ell) capChartDomain ∧
ContDiffOn ℝ ∞ (heightChartC (g0 + eta) z Y ell) capChartDomain ∧
ContDiffOn ℝ ∞ (coordinatePrimitive (heightChartB (g0 + eta) z Y ell)) capChartDomain ∧
ContDiffOn ℝ ∞ (capPullback Y (gaussianCurvature (g0 + eta))) capChartDomain ∧
ContDiffOn ℝ ∞ (heightChartG1 (g0 + eta) z Y) capChartDomain ∧
(∀ p ∈ capChartDomain,
heightG1Lower G Z d c e0 ≤ heightChartG1 (g0 + eta) z Y p ∧
|coordinatePrimitive (heightChartB (g0 + eta) z Y ell) p| ≤ 2*heightBJetBound G Z d c ell ∧
(∀ i : Fin 2, |coordPartial i (coordinatePrimitive (heightChartB (g0 + eta) z Y ell)) p| ≤
heightCapCoefficientBound G Z d c e0 ell) ∧
(∀ i : Fin 2, |coordPartial i (capPullback Y (gaussianCurvature (g0 + eta))) p| ≤
heightCapCurvatureBound G Z d c) ∧
|heightChartA (g0 + eta) z Y p| ≤ heightCapCoefficientBound G Z d c e0 ell ∧
(∀ i : Fin 2, |coordPartial i (heightChartA (g0 + eta) z Y) p| ≤
heightCapCoefficientBound G Z d c e0 ell) ∧
|heightChartB (g0 + eta) z Y ell p| ≤ heightCapCoefficientBound G Z d c e0 ell ∧
|heightChartC (g0 + eta) z Y ell p| ≤ heightCapCoefficientBound G Z d c e0 ell) ∧
∀ edge : ℝ, edge ≤ 0 → ∀ p ∈ capChartDomain, p 1 < edge →
capPullback Y (gaussianCurvature (g0 + eta)) p ≤ c1 * (edge - p 1) →
∀ x y : ℝ,
let A := heightChartA (g0 + eta) z Y
let B := heightChartB (g0 + eta) z Y ell
let C := heightChartC (g0 + eta) z Y ell
let I := coordinatePrimitive B
c2 * directedWeight edge lambda I p * (x^2 + y^2) ≤
multiplierT B (directedM edge lambda I) (directedN edge lambda epsilon I) p * x^2 +
multiplierS A C (directedM edge lambda I) (directedN edge lambda epsilon I) p * y^2 +
multiplierJ A B C (directedM edge lambda I) (directedN edge lambda epsilon I) p * x * y := by
let M := heightQuotientJetBound G Z d c
let cG := heightG1Lower G Z d c e0
let rhoMax := 1 / Real.exp (-2 * M)
let MG := heightG1JetBound G Z d c
let MB := heightBJetBound G Z d c ell
let Mr := heightRemainderBound G Z d c e0 ell
let MA := heightAJetBound G Z d c
have hcG : 0 < cG := heightG1Lower_pos hG hZ hd he0
have hrhoMax : 0 < rhoMax := by dsimp [rhoMax]; positivity
have hMG : 0 ≤ MG := heightG1JetBound_nonneg hG hZ hd hc
have hMB : 0 ≤ MB := heightBJetBound_nonneg hG hZ hd hc ell
have hMr : 0 ≤ Mr := heightRemainderBound_nonneg hG hZ hd hc he0 ell
have hMA : 0 ≤ MA := heightAJetBound_nonneg hG hZ hd hc
let epsilon := heightDirectedEpsilon G Z d c e0 kappa ell
let lambda := heightDirectedLambda G Z d c e0 kappa ell
let c1 := heightDirectedSlope G Z d c e0 kappa ell
obtain ⟨heps, heps1, hlambda, hc1, hqbudget, hTbudget, hEbudget,
hcross, hloss, hfarbudget, hnearstrip, hstripbudget⟩ :=
heightDirectedNumbers_spec ⟨hG,hZ,hd,hc,he0,hk⟩ ell
let margin := modelCoercivityMargin cG kappa rhoMax epsilon ell
let delta := modelPrincipalRadius cG kappa rhoMax epsilon 2 MG ell
let c2 := min (lambda / 8) (margin / 4)
have hmargin : 0 < margin := modelCoercivityMargin_pos ell hcG hk hrhoMax heps
have hdelta : 0 < delta := modelPrincipalRadius_pos ell hcG hk hrhoMax heps (by norm_num) hMG
have hc2 : 0 < c2 := by dsimp [c2]; positivity
obtain ⟨Y, hchart, hYclosed, hYs, hvar, hstart, hode, hconf, hdisp50, hdisp,
hrho, hG1s, hG1B, hGfloor, hall⟩ :=
exists_height_flow_uniform_coefficients hg hU hSU hz hG hZ hd hc he0 hgB hzB hdet hyy hEfloor hsmall hD
have hmapO : MapsTo (capChart Y) capChartDomain modelOpenSquare :=
fun p hp => capChart_mem_modelOpenSquare hp (hdisp p hp)
have hmapS : MapsTo (capChart Y) capChartDomain modelSquare :=
fun p hp => modelOpenSquare_subset (hmapO hp)
have hAB := heightChartA_bound_one hg hU hSU hG hZ hd hc hgB hdet hYs hvar hode hmapS hsmall
(fun p hp => (hrho p hp).2.1) hG1s hG1B
obtain ⟨hBs, hBB, hrs, hrB⟩ := hall ell
have hOU : modelOpenSquare ⊆ U := modelOpenSquare_subset.trans hSU
have hgO : SmoothPositiveOn (g0 + eta) modelOpenSquare :=
⟨fun i j => (hg.1 i j).mono hOU, fun p hp => hg.2 p (hOU hp)⟩
have hzO := hz.mono hOU
have hyyn := LowQuotient.denominator_ne_zero hc hyy
have hyyO := fun p (hp : p ∈ modelOpenSquare) => hyyn p (modelOpenSquare_subset hp)
have hEO := fun p (hp : p ∈ modelOpenSquare) => he0.trans_le (hEfloor p (modelOpenSquare_subset hp))
have hDO := fun p (hp : p ∈ modelOpenSquare) => hD p (modelOpenSquare_subset hp)
have hbackgroundO := fun p (hp : p ∈ modelOpenSquare) => hbackground p (hOU hp)
have hAs := heightChartA_contDiffOn hgO modelOpenSquare_isOpen hzO hyyO hYs hvar hmapO
have hKB := heightCurvaturePullback_bound_one hg hU hSU hG hZ hd hc hgB hdet
hYs hvar hode hmapS hsmall (fun p hp => (hrho p hp).2.1)
obtain ⟨hMF,hMFA,hMFI,hMFB,hMFC⟩ := heightCapCoefficientBound_dominates hG hZ hd hc he0 ell
have hfixed (p : Coord) (hp : p ∈ capChartDomain) :
heightG1Lower G Z d c e0 ≤ heightChartG1 (g0 + eta) z Y p ∧
|coordinatePrimitive (heightChartB (g0 + eta) z Y ell) p| ≤ 2*heightBJetBound G Z d c ell ∧
(∀ i : Fin 2, |coordPartial i (coordinatePrimitive (heightChartB (g0 + eta) z Y ell)) p| ≤
heightCapCoefficientBound G Z d c e0 ell) ∧
(∀ i : Fin 2, |coordPartial i (capPullback Y (gaussianCurvature (g0 + eta))) p| ≤
heightCapCurvatureBound G Z d c) ∧
|heightChartA (g0 + eta) z Y p| ≤ heightCapCoefficientBound G Z d c e0 ell ∧
(∀ i : Fin 2, |coordPartial i (heightChartA (g0 + eta) z Y) p| ≤
heightCapCoefficientBound G Z d c e0 ell) ∧
|heightChartB (g0 + eta) z Y ell p| ≤ heightCapCoefficientBound G Z d c e0 ell ∧
|heightChartC (g0 + eta) z Y ell p| ≤ heightCapCoefficientBound G Z d c e0 ell := by
obtain ⟨hIv,hIi⟩ := heightChartB_primitive_first_bounds ell hMB hBs hBB hp
exact ⟨hGfloor p hp,hIv,fun i => (hIi i).trans hMFI,
fun i => hKB [i] (by norm_num) p hp,
(hAB [] (by norm_num) p hp).trans hMFA,
fun i => (hAB [i] (by norm_num) p hp).trans hMFA,
(hBB [] (by norm_num) p hp).trans hMFB,
(heightChartC_value_bound hG hZ hd hc he0 ell hAB hrB hp).trans hMFC⟩
refine ⟨Y, heps, heps1, hlambda, hc1, hc2, hchart, hYs, hvar, hstart, hode, hdisp,
hAs, hBs, heightChartC_contDiffOn hgO modelOpenSquare_isOpen hzO hDO hyyO hEO hYs hvar hmapO ell,
coordinatePrimitive_contDiffOn hBs,
capPullback_contDiffOn hYs (gaussianCurvature_contDiffOn hg hU) (fun p hp => hSU (hmapS hp)),
hG1s,hfixed, ?_⟩
intro edge hedge0 p hp hedge hKstrip x y
have hs : p 1 ≤ 0 := (le_of_lt hedge).trans hedge0
have hdist : edge - p 1 ≤ 2 := by linarith [hp.2.1]
have hI := (heightChartB_primitive_bounds ell hBs hBB hp).2
have hBpoint : |heightChartB (g0 + eta) z Y ell p| ≤ MB := hBB [] (by norm_num) p hp
have hrpoint : |heightChartRemainder (g0 + eta) z Y ell p| ≤ Mr := (hrB p hp).1
have hApoint : |heightChartA (g0 + eta) z Y p| ≤ MA := hAB [] (by norm_num) p hp
have hAspoint : |coordPartial 1 (heightChartA (g0 + eta) z Y) p| ≤ MA := hAB [1] (by norm_num) p hp
have hAtpoint : |coordPartial 0 (heightChartA (g0 + eta) z Y) p| ≤ MA := hAB [0] (by norm_num) p hp
have hGspoint : |coordPartial 1 (heightChartG1 (g0 + eta) z Y) p| ≤ MG := hG1B [1] (by norm_num) p hp
have hGtpoint : |coordPartial 0 (heightChartG1 (g0 + eta) z Y) p| ≤ MG := hG1B [0] (by norm_num) p hp
have hGmax : heightChartG1 (g0 + eta) z Y p ≤ MG := (le_abs_self _).trans (hG1B [] (by norm_num) p hp)
by_cases hnear : |capPullback Y (gaussianCurvature (g0 + eta)) p| ≤ delta
· exact patch_actual_directed_near_coercivity hgO modelOpenSquare_isOpen hzO hDO hyyO hEO
hYs hode hvar hdisp50 hsmall hmapO hbackgroundO hsupport hcentral hp hs ell
cG rhoMax epsilon MG (2 * MB) MB Mr MA MA MG c1 2 edge lambda hk hcG (hGfloor p hp)
(hrho p hp).2.2 heps heps1 hqbudget hGspoint hGtpoint hnear hedge hlambda hI hBpoint hrpoint
hApoint hAspoint hTbudget hEbudget hcross hloss hGmax hc1.le hdist hKstrip hstripbudget x y
· have hKupper : capPullback Y (gaussianCurvature (g0 + eta)) p ≤ delta :=
hKstrip.trans ((mul_le_mul_of_nonneg_left hdist hc1.le).trans hnearstrip)
have hKnegative : capPullback Y (gaussianCurvature (g0 + eta)) p ≤ -delta := by
by_contra hneg
exact hnear (abs_le.mpr ⟨(lt_of_not_ge hneg).le, hKupper⟩)
have hAfar := heightChartA_negative_of_curvature (g0 + eta) z Y p hcG hdelta (hGfloor p hp) hKnegative
have hloss' :
(4 * (2 * ((ell : ℝ) * MA + MA * Mr + MA * (2 * MB) + 1))^2 * epsilon^2) / lambda ≤
(margin / 2) / 4 := by
calc
_ ≤ margin / 8 := hloss
_ = _ := by ring
have hh := primitive_directed_farnegative_coercivity hBs (by norm_num : (0 : ℝ) < 2) hp
(heightChartA (g0 + eta) z Y) (heightChartC (g0 + eta) z Y ell)
(heightChartRemainder (g0 + eta) z Y ell)
((hAs.contDiffAt (capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp))
edge lambda epsilon ell hedge hlambda heps.le heps1 (hrB p hp).2
(2 * MB) MB Mr MA MA MA (margin / 2) (cG * delta) (by positivity)
hI hBpoint hrpoint hApoint hAspoint hTbudget hEbudget hcross hloss' hAtpoint
(mul_pos hcG hdelta) hAfar hfarbudget x y
have hhalf : (margin / 2) / 2 = margin / 4 := by ring
rw [hhalf] at hh
exact hh
end SmoothLocal.Flow
end
end OAI
|